The Hammers are unhappy that the England striker's ban - a consequence of being sent off against Swansea last week - was not overturned, and have taken their case to an independent arbitral tribunal in the pursuit of "justice".
Co-owner David Gold said on Thursday that his club were seeking "legal redress" over the matter, although a statement on West Ham's official website on Friday morning ruled out any sort of court action.
The independent panel, at which the FA will also be represented, is expected to rule later this afternoon, and if the resolution falls in West Ham's favour it would free main striker Carroll up for the Barclays Premier League game at Aston Villa.
Allardyce firmly believes the Irons were right to take the matter further.
Speaking at Friday's pre-match media conference, Allardyce reflected on the FA's verdict and said: "My reaction was anger and injustice.
"I think the whole procedure in terms of how we put our case together and the vast swell of people felt it was unjust and for me the panel has not seen it how they should have seen it and as the evidence we gave.
"It's easy to hide behind the regulations or what the law says, but that's an easy way out for me.
"Referees often say technically it's this and that or 'the letter of the law' and that and it grinds on you.
"In this case they were looking at it from the view of one thing only: was it an obvious mistake? So we based our procedure on this, and I'm 100 per cent certain it was an obvious mistake?
"(Referee) Howard Webb should have given a free-kick for Andy against (Chico) Flores, at that stage the whistle blows and there's no incident, so that's an obvious mistake, it's an obvious free-kick.
"The second thing is: did he have a clear view and clear eye line of the incident? Howard had just watched it in his dressing room when I went to talk to him. And so that's a great piece of evidence to say, 'Did you see clearly what had happened?'.
"My next question was, 'Who made the decision?'. His answer was he did and he didn't use assistance from fourth official or the assistant referees.
"So we based our whole appeal on that scenario and for me the conclusion could only have been that he felt that even though he hadn't seen it 100 per cent, he was reluctant not to give a red card on the basis that if Andy had caught him full in the face or elbowed him or used violent conduct, which he didn't, it was at very best reckless, then he would've been in trouble with his group of referees and his bosses for not giving it."
Allardyce was left less than impressed by the reaction of Spanish defender Flores.
The West Ham manager said: "The other scenario is that simulation is allowed to win. Mr Flores seems to be pretty proud of himself Tweeting what has happened in that scenario, but he has got away with simulating.
"He has caused a bigger problem in football than what the arbitration panel is looking at for me.
"He has allowed people to say to players 'you can get away with simulation to get someone sent off or get a decision in your favour', that is clear for all to see on what happened.
"Howard thought that Andy has smashed him straight in the face by his reaction and then when you see it after there is little or no contact and it is nowhere near his face, so it is a shame.
"The other scenario that he must feel a lot more guilty of is he has just got his manager sacked."
Allardyce presented some statistics to the media ahead of the conference, highlighting an increase in the number of red cards shown this season.
He added: "The stats about the increase in sending-offs in 2014 suggest the referees have been told they must give more red cards than they've been giving previously.
"That proves there's an increase and the only conclusion I can come to is that they've been told they haven't been giving enough red cards out.
"The sad thing is if you're giving out red cards you have to be 100 per cent certain. We've had two and they couldn't have been 100 per cent certain."
Allardyce expects his players to provide a positive response at Villa Park.
He said: "We all have to overcome adversity. I hope the players are angry, as angry as I am, and I hope they take that angry out in a positive manner against Aston Villa to go and prove we can still win a game of football without Andy."